Groupoid

From Online Dictionary of Crystallography

A groupoid (G,*) is a set G with a law of composition * mapping of a subset of G x G into G. The properties of a groupoid are:

if x, y, z ∈ G and if one of the compositions (x*y)*z or x*(y*z) is defined, so is the other and they are equal; (associativity);

if x, x' and y ∈ G are such that x*y and x'*y are defined and equal, then x = x'; (cancellation property)

for all x ∈ G there exist elements ex (left unit of x), ex' (right unit of x) and x-1 ("inverse" of x) such that:

ex*x = x

x* ex' = x

x-1*x = ex'.

From these properties it follows that:

x* x-1 = ex, i.e. that that ex is right unit for x-1,

ex' is left unit for x-1

ex and ex' are idempotents, i.e. ex* ex = ex and ex'* ex' = ex'.

The concept of groupoid as defined here was introduced by Brandt (1927). An alternative meaning of groupoid was introduced by Hausmann & Ore (1937) as a set on which binary operations act but neither the identity nor the inversion are included. For this second meaning nowadays the term magma is used instead (Bourbaki, 1998).